COMEDK · Maths · 28. Indefinite Integration
\(\text { The value of } \int \dfrac{d x}{\sqrt{2 x-x^2}} \text { is }\)
- A \(\sin ^{-1}(2 x-1)+C\)
- B \(\sin ^{-1}(x+1)+C\)
- C \(-\sqrt{2 x-x^2}+C\)
- D \(\sin ^{-1}(x-1)+C\)
Answer & Solution
Correct Answer
(D) \(\sin ^{-1}(x-1)+C\)
Step-by-step Solution
Detailed explanation
The integral is \(I = \int \dfrac{dx}{\sqrt{2x - x^2}}\). Complete the square for the expression inside the square root: \(2x - x^2 = -(x^2 - 2x) = -(x^2 - 2x + 1 - 1) = 1 - (x - 1)^2\). Substituting this into the integral: \(I = \int \dfrac{dx}{\sqrt{1 - (x - 1)^2}}\). Using…
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